15.20.8 problem 8
Internal
problem
ID
[3316]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
38,
page
173
Problem
number
:
8
Date
solved
:
Tuesday, March 04, 2025 at 04:34:30 PM
CAS
classification
:
[_dAlembert]
\begin{align*} x&=y^{\prime } y+{y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.034 (sec). Leaf size: 304
ode:=x = y(x)*diff(y(x),x)+diff(y(x),x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\frac {\left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) c_{1}}{\sqrt {-2 y \left (x \right )+2 \sqrt {y \left (x \right )^{2}+4 x}-4}\, \sqrt {-2 y \left (x \right )+2 \sqrt {y \left (x \right )^{2}+4 x}+4}}+x +\frac {\left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) \left (-\ln \left (2\right )+\ln \left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}+\sqrt {2 y \left (x \right )^{2}-2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}\right )\right )}{\sqrt {2 y \left (x \right )^{2}-2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}} &= 0 \\
\frac {\left (y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) c_{1}}{\sqrt {-2 y \left (x \right )-2 \sqrt {y \left (x \right )^{2}+4 x}-4}\, \sqrt {-2 y \left (x \right )-2 \sqrt {y \left (x \right )^{2}+4 x}+4}}+x -\frac {\left (y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) \left (-\ln \left (2\right )+\ln \left (-y \left (x \right )-\sqrt {y \left (x \right )^{2}+4 x}+\sqrt {2 y \left (x \right )^{2}+2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}\right )\right )}{\sqrt {2 y \left (x \right )^{2}+2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.645 (sec). Leaf size: 59
ode=x==D[y[x],x]*y[x]+D[y[x],x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left \{x=\frac {K[1] \arcsin (K[1])}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}-K[1]\right \},\{y(x),K[1]\}\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x - y(x)*Derivative(y(x), x) - Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(4*x + y(x)**2)/2 + y(x)/2 + Derivative(y(x), x) cannot be solved by the factorable group method